Removing Parentheses

Sometimes we need to group several terms together, and sometimes that group sits inside another group. To keep them apart we use parentheses (\;), brackets [\;], and braces \{\;\}. Together they're called signs of aggregation, and people often just call them all "parentheses."

A plus sign in front

12 + (5 - 3) is the same as 12 + 5 - 3, which is 14. Likewise a + (b - c) = a + b - c. The plus sign in front belongs to the parenthesis and disappears with it; the terms inside keep their own signs. So 12 + (-5 - 3) = 12 - 5 - 3 = 4.

Principle. A parenthesis preceded by a plus sign may be removed without changing the signs of the terms inside.

A minus sign in front

12 - (5 - 3) means subtract the binomial 5 - 3 from 12, so we change the signs of what's being subtracted: 12 - 5 + 3 = 10. That's clearly right, since 12 - 2 = 10. Likewise a - (b - c) = a - b + c.

Principle. A parenthesis preceded by a minus sign may be removed if the sign of every term inside is changed.

Parentheses inside parentheses

Rule. Rewrite the expression without the innermost parenthesis, changing the signs inside if a minus sign precedes it. Combine any like terms in the new innermost group. Repeat until every parenthesis is gone.

\begin{aligned} 5a - [2a + (-3a - 4b) - (a - 3b) + 4a] &= 5a - [2a - 3a - 4b - a + 3b + 4a] \\ &= 5a - [2a - b] \\ &= 3a + b \end{aligned}

Typing tip: you can type brackets and braces, or just use ordinary parentheses.